Concerning road traffic, the research activity during 2011 focused on the mathematical analysis of traffic flow models on road networks or subject to unilateral constraints. In particular, is devoted to a hyperbolic 2nd order model for traffic flow with local flux constraint. We describe two admissible Riemann solvers and we construct ad hoc finite volume numerical schemes to compute these solutions. The paper is devoted to the study of a traffic flow model on a network composed by an arbitrary number of incoming and outgoing arcs connected together by a node with a buffer. We define the solution to the Riemann problem at the node and we prove existence and well posedness of solutions to the Cauchy problem. Finally, a general traffic flow model with phase-transition is proposed and described in .

M.L. Delle Monache just started her doctoral thesis in the same topic. More precisely she will study hyperbolic models of traffic flow and associated optimization problems.

Concerning pedestrian motion modeling, we are interested in the optimization of facilities design, in order to maximize pedestrian flow and avoid or limit accidents due to panic situations. To this aim, we are now studying a macroscopic model for crowd movements consisting in a scalar conservation law accounting for mass conservation coupled with an Eikonal equation giving the flux direction depending on the density distribution. From the theoretical point of view, and as a first step, we are studying the problem in one space dimension (for applications, this case corresponds to a crowd moving in a corridor). In collaboration with M.  Rosini (supported by the project CROM3, funded by the PHC Polonium 2011), we have established entropy conditions to select physically relevant solutions, and we have constructed explicit solutions for some simple initial data (these results are presented in  ). We are now studying existence and uniqueness of solutions of the corresponding initial boundary value problem. From the numerical point of view, we are implementing the model in two space dimensions on triangular meshes on the Num3sis platform. This was partly done by N. El-Khatib (postdoc at INRIA from January to August 2011), and will be completed soon by Nora Aïssiouene. This will provide a performing numerical tool to solve the related optimization problems arising in the optimization of facilities design, such as the position and size of an obstacle in front of (before) a building exit in order to maximize the outflow through the door and avoid or limit over-compression. Moreover, jointly with the PULSAR team, we have supervised J. Joobeur's internship, which was devoted to pedestrian data collection from real-word video recordings (Turin metro station). The density data will serve to validate the model.

The above researches were partially funded by the ERC Starting Grant "TRAM3 - Traffic management by macroscopic models".

The framework of our research aims to contribute to numerical strategies for PDE-constrained multiobjective optimization, with a particular emphasis on CPU-demanding computational applications in which the different criteria to be minimized (or reduced) originate from different physical disciplines that share the same set of design variables. These disciplines are often fluids, as a primary focus, coupled with some other discipline, such as structural mechanics.

Our approach to competitive optimizationis based on a particular construction of Nash games, relying on a split of territoryin the assignment of individual strategies. A methodology has been proposed for the treatment of two-discipline optimization problems in which one discipline, the primary discipline, is preponderant, or fragile. Then, it is recommended to identify, in a first step, the optimum of this discipline alone using the whole set of design variables. Then, an orthogonal basis is constructed based on the evaluation at convergence of the Hessian matrix of the primary criterion and constraint gradients. This basis is used to split the working design space into two supplementary subspaces to be assigned, in a second step, to two virtual players in competition in an adapted Nash game, devised to reduce a secondary criterion while causing the least degradation to the first. The formulation has been proved to potentially provide a set of Nash equilibrium solutions originating from the original single-discipline optimum point by smooth continuation, thus introducing competition gradually. This approach has been demonstrated over a testcase of aero-structural aircraft wing shape optimization, in which the eigen-split-based optimization reveals clearly superior .

While the two-discipline method is currently being applied to various complex physical multiobjective situations (see in particular , , , ), the method has been extended to situations involving more than two objectives when the initial point is Pareto-optimal. Then, a particular convex combination of the criteria is locally stationary, and the two-discipline strategy can be applied using this combination as preponderant criterion, and a particular other criterion as secondary one. Whence, the proposed split of territory produces a continuum of Nash equilibrium points tangentto the Pareto set. This theoretical result has been illustrated in the context of a simpler numerical experiment by E. Baratchart during his internship , see Fig.  .

Our approach to cooperative optimizationis based on a result of convex analysis established for a general unconstrained multiobjective problem in which all the gradients are assumed to be known. The theorem states that in the convex hull of the gradients, there exists a unique vector of minimal norm, $\omega$; if it is nonzero, the vector $\omega$is a descent direction common to all criteria; otherwise, the current design point is Pareto-optimal. This result led us to generalize the classical steepest-descent algorithm by using the vector $\omega$as search direction. We refer to the new algorithm as the multiple-gradient descent algorithm (MGDA). The MGDA yields to a point on the Pareto set, at which a competitive optimization phase can possibly be launched on the basis of the local eigenstructure of the different Hessian matrices. This general formulation fosters several connected studies detailed in .

In several engineering problems, the system to optimize is characterized by some parameters that define global shape properties, while remaining parameters define local shape modifications. Of course, these two sets of parameters do not play the same role and have not the same impact on the cost functional value. Therefore, we are studying how to construct an efficient optimization strategy that takes benefit of this global / local splitting of parameters.

A typical aerodynamic shape optimization problem has been studied, that consists of a lift-constrained drag minimization for a transonic wing, whose sections are defined by two B-Spline curves whereas global shape characteristics are defined by five parameters (span, roottip length ratio, angle of attack, twist angle, sweep angle). It has been found that the naive simultaneous optimization of all parameters failed, due to the multimodality of the problem. Alternatively, the use of a virtual game strategy, based on a splitting between the local and global parameters, yields a satisfactory result for a moderate cost  .

Shape optimization methods are not efficient to improve the performance of fluid systems, when the flow is characterized by a strong unsteadiness related to a massive flow separation. This is typically the case for the flow around an automotive body or a wing in stall condition. To overcome this difficulty, flow control strategies are developed, that aim at manipulating vortex dynamics by introducing some active actuators, such as periodic blowing/suction jets. In this context, the choice of the control parameters (location, amplitude, frequency) is critical and not straightforward. Therefore, a numerical study is conducted to i) improve the understanding of controlled flows ii) develop a methodology to determine optimal control parameters by coupling the controlled flow simulation with optimization algorithms. Two research axes have been considered :

At ONERA, compressible flow simulations governed by the Euler or Navier Stokes (RANS) equations are conducted with the software elsA that admits both structured and unstructured-grid formulations. Local aerodynamic optimizations are made with a version that includes the calculation of the shape gradient via the solution of an adjoint equation. The discrete adjoint is calculated formally step-by-step to include the various derivative terms involved, and is being enhanced gradually to account for more complex models. In particular, for RANS computations, this gradient today includes the differentiation of the turbulence model.

In industry, aircraft wings are designed by accounting for several multidisciplinary couplings. Certainly of greatest importance is the coupling, or concurrency, between aerodynamic optimization and structural design. At ONERA, in the former thesis of M. Marcelet, the aerodynamic gradient has been extended to account for (the main terms of) static fluid-structure interaction, commonly referred to as the “aeroelastic gradient”.

In her thesis, I. Ghazlane has extended M. Marcelet's work to take into account, in the aeroelastic gradient, the terms originating from the differentiation of the wing-structural model. In this development, the wing structure is treated as an equivalent Euler-Bernoulli beam. These formal extensions have been validated by an extensive experimentation. Additionally, special post-processing procedures are applied to evaluate accurately the various physical contributions to drag. As a result, the numerical tools necessary to conduct a very realistic aircraft wing optimization are now set up and are being exploited . It is also envisaged to conduct a two-objective optimization (drag and mass reduction) via a Nash game using our optimization platform FAMOSA.

When an aircraft flies at supersonic speed, it generates at ground level an N-shaped shock structure which can cause serious environmental damage (“sonic boom”). Thus a problem of interest in aerodynamic optimization is to design such an aircraft to reduce the intensity of the sonic boom while maintaining the aerodynamic performance (drag minimization under lift constraint). Andrea Minelli's aimed at contributing to this two-discipline optimization problem. In the first part of his work, an inverse problem has been formulated and solved for “shaped sonic boom” and found in excellent agreement with the George-Seebass-Darden theory for the calculation of the Whitham function corresponding to the lowest-boom (axisymmetric) shape. The method is currently being extended to account for more general geometries. Besides, aero-acoustic optimizations have been realized successfully by coupling the aerodynamic optimizer (based on Euler calculations by the elsA sofware) with the sonic-boom computation in a Nash game formulation. These experiments, conducted with our optimization platform FAMOSA, have demonstrated that starting from the shape optimized aerodynamically, one could retrieve smoothly a shape corresponding to nearly-optimal sonic-boom reduction.

E. Roca Leon has recently started at ONERA a CIFRE thesis supported by EUROCOPTER, Marignane. This thesis follows the doctoral thesis of A. Dumont in which the adjoint-equation approach was used to optimize a rotor blade in hovering flight. The goal of this new thesis is to solve a two-objective optimization problem in which the hovering-flight criterion is considered preponderant, but a new criterion that takes into account the forward-flight situation is also introduced, concurrently. The thesis work includes the set up of a hierarchy of models from low to high fidelity, in order to calibrate appropriate functional criteria. Secondly, our Nash game approach to competitive optimization will be implemented, using our optimization platform FAMOSA, and comparisons with the results by A. Dumont will be made.

Naval hydrodynamics field has recently shown a growing interest for optimum design methods. The computational context is especially complex because it implies unsteady two-phase turbulent flows, with possibly very high Reynolds number (up to ${10}^9$). The use of automated design optimization methods for such problems requires new developments to take into account the large CPU time necessary for each simulation and the specificity of the geometries considered.

In collaboration with GALAAD Project-Team, some developments have been initiated on the geometrical modeling of hull shapes by parametric surfaces. The objective is to be able to modify existing hull shapes by controlling a small number of parameters, that are meaningful for naval architects. Two testcases are considered: the bow shape for trawler ships (see Fig.  ) and the whole hull shape for canoes, in collaboration with the Fédération francaise de Canoe-Kayak.

In collaboration with the ArcelorMittal's Center for Research in Automotive and Applications, we study the multidisciplinary shape and parameter design of highly non linear mechanical 2D and 3D structures. We have developed methods adapted to the approximation of Pareto Fronts such as Normal Boundary Intersection NBI and Normalized Normal Constraint Method NNCM. Due to the time consuming cost evaluation, the use of cheap to evaluate surrogate models is mandatory. We have studied the consistency of the approach NBI or NNCM plus surrogates, which turned out to be successful for a broad panel of standard mathematical benchmarks. The application of this approach for the case of beverage cans which undergo elastoplastic deformation under high pressure is ongoing.

Within the OASIS Consortium (ArcelorMittal, ErDF, INRIA, UTC, EURODECISION, ESILV, NECS, DeltaCAD, SCILAB-DIGITEO), Opale Project leads the Optimization task. Our aim is to develop decentralized decision-making algorithms dedicated to find efficient solutions (Pareto optimal) in a complex multidisciplinary framework (forming, stamping, welding non-linear processes, spring-back, vibration, in-function linear processes, crash and fatigue non linear and non differentiable processes) for several (between three and five) criteria. An important difficulty when trying to identify the Pareto Front, even when using adapted methods such the Normal Boundary Intersection, is that the criteria involved (thanks to the high nonlinearity in the mechanical models) exhibit many local optima. So one must use global optimization methods. We have studied the hybrid approach Simulated Annealing with Simultaneous Perturbation SASP for a suite of mathematical test-cases. To envisage the application of our method to the complex CPU time consuming stamping process, we lead an intermediate phase dedicated to the validation of the SASP method for the minimization of the spring-back that follows the stamping of a metal sheet, the design variable being the thickness distribution.

A mathematical model for simulation of actin cable contraction, during wound closure for Drosophila embryo, which contains an extra term in addition to the curvature flow is developed. The basic mathematical model introduced and validated in is extended in order to include the non-homogeneous wound healing or non-homogeneous dorsal closure .

Similar to the discretization of ordinary or partial differential equations, the numerical approximation of the solution of an optimization problem is possibly subject to numerical stiffness. In the framework of parametric shape optimization, hierarchical representations of the shape can be used for preconditioning, following the idea of Multigrid (MG) methods. By analogy with the Poisson equation, which is the typical example for linear MG methods, we have addressed a parametric shape inverse problem. The ideal cycle of a two-level algorithm can be defined and adapted to shape optimization problems that require appropriate transfer operators. With the help of a symbolic calculus software we have shown that the efficiency of an optimization MG-like strategy is ensured by a small dimension-independent convergence rate. Numerical examples are worked out and corroborate the theoretical results. Applications to antenna design have been realized. Finally, some connections with the direct and inverse Broyden-Fletcher-Goldfarb-Shanno preconditioning methods have been shown .

M. Nguyen Dinh is conducting a CIFRE thesis at ONERA supported by AIRBUS France. The thesis topic is the qualification of CFD simulations by anisotropic mesh adaption. Methods for refining the 2D or 3D structured mesh by node movement have been examined closely. Secondly, it is investigated how could the local information on the functional gradient $∥dJ/dX∥$be exploited in a multi-block mesh context. This raises particular questions related to conservation at the interfaces.

S. Bourasseau has started a CIFRE thesis at ONERA supported by SNECMA. The thesis is on mesh adaption in the context of hybrid meshes, that is, made of both structured and unstructured regions. Again, the aim is to exploit at best the function gradient provided by the adjoint-equation approach. Preliminary experiments have been conducted on geometries of stator blade yielding the sensitivities to global shape parameters.

We have started in 2011 to study the application of game modeling to image processing problems. We propose an original game theory approach to simultaneously restore and segment noisy images . We define two players: one is restoration, with the image intensity as strategy, and the other is segmentation with contours as strategy. Cost functions are the classical relevant ones for restoration and segmentation, respectively. The two players play a static game with complete information, and we consider as solution to the game the so-called Nash Equilibrium. For the computation of this equilibrium we present an iterative method with relaxation. The results of numerical experiments performed on some real images show the relevance and efficiency of the proposed algorithm. Based on a similar idea, we formulated well known data completion (Cauchy) problems for Laplace equation as Nash games and obtained results of existence, uniqueness and stability of a Nash equilibrium which turns out to be the Cauchy solution when the Cauchy data are compatible. With A. Theljani, we study the extension of the Nash data completion approach to nonlinear parabolic equations with application to image impainting.